How to create and export multiple animation/actions in blender?
Wavefront OBJ is a widely supported file format for 3D models. It can store vertex coordinates, UVs, vertex normals, face definitions and texture vertices (plus additional, but not standard-compliant data in some softwares).

Animations are not supported. The export option Animation in the OBj exporter does not really export an animation, but an OBJ for every frame in your scene. So it's really a series of OBJ models. Some engines may support this for animations.

As said, the addon uses the scene, so the current timeline. Every object can reference not more than a single action. Therefore, only that one action is exported. Batch export of all actions is not supported.

It would be trivial to do code-wise, but we lack an essential information: which action belongs to which object? All actions are stored globally (bpy.data.actions in python). For which objects an action works, is not stored. You can "run" an action on any object, e.g. matching bones in another, different armature will be animated too.

Thus, you need to switch actions on your animated object, change start and end frame on your timeline and export them one by one. Or adapt the export addon to batch-export all actions for the active object (you may use Action.frame_range to determine the keyframed boundaries).

You can check the export by either inspecting the output folder, or by checking the system console (on Windows, it's as easy as Window > Toggle System Console):

OBJ Export path: 'C:\\Users\\CoDEmanX\\Desktop\\untitled_000249.obj'
OBJ Export time: 0.00
OBJ Export path: 'C:\\Users\\CoDEmanX\\Desktop\\untitled_000250.obj'
OBJ Export time: 0.00